Solve the following intial value problem

1. The problem statement, all variables and given/known data
Solve the following inital value problem

y1' = y2
y2' = -5y1-4y2

with inital conditions y1(0) = 1, y2(0) = 0

2. Relevant equations

(A-[itex]\lambda[/itex]I)x = 0

3. The attempt at a solution
So I first started by setting out like this

y2' = -5y1 - 4y2
which then suggest 0 = -[itex]\lambda[/itex](-4-[itex]\lambda[/itex]) + 5
0 = [itex]\lambda[/itex]2+4[itex]\lambda[/itex] + 5
But this is going to give me complex roots.. so I was just wondering have I made a mistake in my characteristic equation?1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution1. The problem statement, all variables and given/known data

\begin{equation}y_1^{\prime} = y_2 \end{equation} implies that \begin{equation}y_1^{\prime \prime} = y_2^{\prime} \end{equation}
The second equation you gave can thus be rewritten in terms of just y_1 and its derivatives,
\begin{equation}y_2^{\prime} = y_1^{\prime \prime}= -5y_1 - 4y_1^{\prime} \rightarrow y_1^{\prime \prime} + 4y_1^{\prime}+5y_1 = 0\end{equation}
You just have to solve the equation. Think of the physical parts of the equation. This should be decaying exponentially, but oscillating as well. Try guessing something like
\begin{equation}
y_1(t) = e^{-2t}(c_1 \sin(t) + c_2 \cos(t))
\end{equation}
Use your initial conditions to determine the constants.